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Register a new userLog surfaces of Picard rank one from four lines in the plane
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We derive simple formulas for the basic numerical invariants of a singular surface with Picard number one obtained by blowups and contractions of the four-line configuration in the plane. As an application, we establish the smallest positive volume and the smallest accumulation point of volumes of log canonical surfaces obtained in this way.
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We classify rank two vector bundles on a del Pezzo threefold $X$ of Picard rank one whose projectivizations are weak Fano. We also investigate the moduli spaces of such vector bundles when $X$ is of degree five, especially whether it is smooth, irreducible, or fine.
Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Iltens method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.
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Padmavathi Srinivasan
, Kirsten Wickelgren (with an appendix byn Borys Kadets
, Padmavathi Srinivasan
2018
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in $mathbb{A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over $k$ is relaxed to the condition that the set of four lines being defined over $k$.
Let $M$ be a complex manifold. We prove that a compact submanifold $Ssubset M$ with splitting tangent sequence (called a splitting submanifold) is rational homogeneous when $M$ is in a large class of rational homogeneous spaces of Picard number one. Moreover, when $M$ is irreducible Hermitian symmetric, we prove that $S$ must be also Hermitian symmetric. The basic tool we use is the restriction and projection map $pi$ of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with $dimgeq 2$ in a hyperquadric, which has been previously proven using algebraic geometry.
Let $mathcal Csubset(0,1]$ be a set satisfying the descending chain condition. We show that any accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $mathcal C$ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and coefficients in $overline{mathcal C}cup{1}$, with at least one coefficient in $Acc(mathcal C)cup{1}$. As a corollary, if $overline{mathcal C}subsetmathbb Q$ then all accumulation points of volumes are rational numbers, solving a conjecture of Blache. For the set of standard coefficients $mathcal C_2={1-frac{1}{n}mid ninmathbb N}cup{1}$ we prove that the minimal accumulation point is between $frac1{7^2cdot 42^2}$ and $frac1{42^2}$.
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